SOLUTION: I am given the expression: {{{ sec(x)*((sin(x))/(tan(x))) }}} Which of the following expressions is the above expression NOT equal to? A. {{{ csc^(2)(x)-cot^(2)(x) }}} B. {

Hi, there--

THE PROBLEM:
I am given the expression:


Which of the following expressions is the above expression NOT equal to?
A. 
B. 
C. 
D. 1
E. 

I *think* I've narrowed it down to either A, C or E, but am not sure. Thank you for your help!

A SOLUTION:

SOME IDEAS
One strategy is to rewrite the expressions in terms of sin(x) and cos(x). This can make it easier to see 
what you have going on. Let's try that with .



Now simplify. Change division by sin(x)/cos(x) to multiplication by its reciprocal.


Now we can cancel out sin(x) and cos(x) in the numerator and denominator.


We see that 


Look! We have already eliminated answer [D].

Let's try this with answer [A].


Rewrite the right-hand side as one fraction with a denominator of sin^2(x).


A trig identity that comes in handy ALL the time is sin^2(x) + cos^2(x) = 1. We can rewrite this identity
as sin^2(x) = 1 - cos^2(x).

Substitute sin^2(x) for 1 - cos^2(x) in the numerator.


We see that this expression is also equal to 1. We have eliminated answer [A].



You can try this technique with answer [C]. 

For answer [E]: 
We know that our original expression is equal to 1. Recall that cos^2(x) + sin^2(x) = 1. 
If we can show that cos^(2)(x) - sin^(2)(x) is not equal cos^(x) + sin^2(x), we will know that answer 
[E] is not equivalent to the original expression. 

Suppose that .

Subtract cos^2(x) from both sides.


Subtract sin^2(x) from both sides of the equation.


Divide both sides by -2.


Take the square root of both sides.


This tells us that  and  are only equal when sin(x) = 0.
Therefore, answer [E] is not equivalent to the original expression for all values of x.

That leaves answer [B] to check. If you graph the expression, you will see that it is not equal to 1 for all values of x. 
However, proving that algebraically is a little messy. If you are interested in seeing the math. Email me and I'll add it here.

Hope this helps! Feel free to email if you have ANY questions about the solution.

Good luck with your math,

Mrs. F
math.in.the.vortex@gmail.com